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-数学- sin(n*%pi/11)^2を解に持つ5次方程式を解く

 

Saito munetakaさんから頂いたコメントの中である5次方程式を解いてみてほしいとのご依頼がありました。早速やってみたところサクッと解けたので記事してみます。ただし、ここで求めた解が\( \left(\sin{\frac{n\,\pi}{11}}\right)^{2} \)に一致することを数値計算ではなく、代数計算で示すのはやっていません。

 

頂いた方程式は(%o2)の式です。

(%i2) (-1024*t^5)+2816*t^4-2816*t^3+1232*t^2-220*t+11;
$$ \tag{%o2} -1024\,t^5+2816\,t^4-2816\,t^3+1232\,t^2-220\,t+11 $$

この式はモニックではないので、このままではSolveSolvable()では解けません。しかしこの場合非常に簡単な変数変換でモニックできます。

(%i3) %,t:-x/4;
$$ \tag{%o3} x^5+11\,x^4+44\,x^3+77\,x^2+55\,x+11 $$

では早速この5次方程式を解いてみます。
(%i4) SolveSolvable(%)$
$$ \tag{*} \verb|Minimal polynomial of V|\verb| | \\
V^5+44\,V^4+572\,V^3+979\,V^2-16500\,V-7249\verb| | $$
$$ \tag{*} \verb|Galois Group of|\verb| |x^5+11\,x^4+44\,x^3+77\,x^2+55\,x+11\verb| | \\
\begin{pmatrix} a&b&c&d&e\\ b&d&e&c&a\\ c&e&b&a&d\\ d&c&a&e&b\\ e&a&d&b&c\\ \end{pmatrix} \verb| | $$
$$ \tag{*} \verb|Subnormal series and quotients of orders|\verb| | \\
\verb|FiniteGroup[1,42,69,86,101]|\verb| | \\
\verb|FiniteGroup[1]|\verb| | \\
x^5+11\,x^4+44\,x^3+77\,x^2+55\,x+11\verb| |\verb|is solvable.|\verb| | $$
$$ \tag{*} \verb|Solutions|\verb| | $$
$$ \tag{*} \mathrm{\%x}_{1}=\frac{589858974750\,\alpha_{1}^4\,{\rm Z}_{5}^3}{24528751927801}-\frac{90453650\,\alpha_{1}^3\,{\rm Z}_{5}^3}{3323228821}-\frac{258355\,\alpha_{1}^2\,{\rm Z}_{5}^3}{4952651}+\frac{53\,\alpha_{1}\,{\rm Z}_{5}^3}{671}+\frac{583898166000\,\alpha_{1}^4\,{\rm Z}_{5}^2}{24528751927801}-\frac{151156000\,\alpha_{1}^3\,{\rm Z}_{5}^2}{3323228821}-\frac{413170\,\alpha_{1}^2\,{\rm Z}_{5}^2}{4952651}+\frac{164\,\alpha_{1}\,{\rm Z}_{5}^2}{671}-\frac{8139626750\,\alpha_{1}^4\,{\rm Z}_{5}}{24528751927801}+\frac{40229400\,\alpha_{1}^3\,{\rm Z}_{5}}{3323228821}+\frac{70770\,\alpha_{1}^2\,{\rm Z}_{5}}{4952651}+\frac{4\,\alpha_{1}\,{\rm Z}_{5}}{671}+\frac{948189648375\,\alpha_{1}^4}{24528751927801}-\frac{217360475\,\alpha_{1}^3}{3323228821}-\frac{650800\,\alpha_{1}^2}{4952651}+\frac{150\,\alpha_{1}}{671}-\frac{11}{5}\verb| | $$
$$ \tag{*} \mathrm{\%x}_{2}=-\frac{5960808750\,\alpha_{1}^4\,{\rm Z}_{5}^3}{24528751927801}+\frac{191385400\,\alpha_{1}^3\,{\rm Z}_{5}^3}{3323228821}-\frac{721570\,\alpha_{1}^2\,{\rm Z}_{5}^3}{4952651}-\frac{150\,\alpha_{1}\,{\rm Z}_{5}^3}{671}-\frac{597998601500\,\alpha_{1}^4\,{\rm Z}_{5}^2}{24528751927801}-\frac{66204475\,\alpha_{1}^3\,{\rm Z}_{5}^2}{3323228821}-\frac{70770\,\alpha_{1}^2\,{\rm Z}_{5}^2}{4952651}-\frac{97\,\alpha_{1}\,{\rm Z}_{5}^2}{671}+\frac{358330673625\,\alpha_{1}^4\,{\rm Z}_{5}}{24528751927801}+\frac{151156000\,\alpha_{1}^3\,{\rm Z}_{5}}{3323228821}-\frac{329125\,\alpha_{1}^2\,{\rm Z}_{5}}{4952651}+\frac{14\,\alpha_{1}\,{\rm Z}_{5}}{671}-\frac{589858974750\,\alpha_{1}^4}{24528751927801}+\frac{60702350\,\alpha_{1}^3}{3323228821}-\frac{483940\,\alpha_{1}^2}{4952651}-\frac{146\,\alpha_{1}}{671}-\frac{11}{5}\verb| | $$
$$ \tag{*} \mathrm{\%x}_{3}=\frac{956329275125\,\alpha_{1}^4\,{\rm Z}_{5}^3}{24528751927801}-\frac{60702350\,\alpha_{1}^3\,{\rm Z}_{5}^3}{3323228821}+\frac{650800\,\alpha_{1}^2\,{\rm Z}_{5}^3}{4952651}-\frac{160\,\alpha_{1}\,{\rm Z}_{5}^3}{671}+\frac{8139626750\,\alpha_{1}^4\,{\rm Z}_{5}^2}{24528751927801}+\frac{130683050\,\alpha_{1}^3\,{\rm Z}_{5}^2}{3323228821}+\frac{392445\,\alpha_{1}^2\,{\rm Z}_{5}^2}{4952651}-\frac{14\,\alpha_{1}\,{\rm Z}_{5}^2}{671}+\frac{597998601500\,\alpha_{1}^4\,{\rm Z}_{5}}{24528751927801}-\frac{126906825\,\alpha_{1}^3\,{\rm Z}_{5}}{3323228821}+\frac{237630\,\alpha_{1}^2\,{\rm Z}_{5}}{4952651}-\frac{164\,\alpha_{1}\,{\rm Z}_{5}}{671}+\frac{592037792750\,\alpha_{1}^4}{24528751927801}+\frac{90453650\,\alpha_{1}^3}{3323228821}+\frac{721570\,\alpha_{1}^2}{4952651}-\frac{111\,\alpha_{1}}{671}-\frac{11}{5}\verb| | $$
$$ \tag{*} \mathrm{\%x}_{4}=-\frac{592037792750\,\alpha_{1}^4\,{\rm Z}_{5}^3}{24528751927801}+\frac{217360475\,\alpha_{1}^3\,{\rm Z}_{5}^3}{3323228821}-\frac{154815\,\alpha_{1}^2\,{\rm Z}_{5}^3}{4952651}+\frac{146\,\alpha_{1}\,{\rm Z}_{5}^3}{671}+\frac{364291482375\,\alpha_{1}^4\,{\rm Z}_{5}^2}{24528751927801}+\frac{126906825\,\alpha_{1}^3\,{\rm Z}_{5}^2}{3323228821}+\frac{329125\,\alpha_{1}^2\,{\rm Z}_{5}^2}{4952651}-\frac{4\,\alpha_{1}\,{\rm Z}_{5}^2}{671}-\frac{583898166000\,\alpha_{1}^4\,{\rm Z}_{5}}{24528751927801}+\frac{66204475\,\alpha_{1}^3\,{\rm Z}_{5}}{3323228821}-\frac{392445\,\alpha_{1}^2\,{\rm Z}_{5}}{4952651}+\frac{49\,\alpha_{1}\,{\rm Z}_{5}}{671}+\frac{5960808750\,\alpha_{1}^4}{24528751927801}+\frac{257589875\,\alpha_{1}^3}{3323228821}+\frac{258355\,\alpha_{1}^2}{4952651}+\frac{160\,\alpha_{1}}{671}-\frac{11}{5}\verb| | $$
$$ \tag{*} \mathrm{\%x}_{5}=-\frac{948189648375\,\alpha_{1}^4\,{\rm Z}_{5}^3}{24528751927801}-\frac{257589875\,\alpha_{1}^3\,{\rm Z}_{5}^3}{3323228821}+\frac{483940\,\alpha_{1}^2\,{\rm Z}_{5}^3}{4952651}+\frac{111\,\alpha_{1}\,{\rm Z}_{5}^3}{671}-\frac{358330673625\,\alpha_{1}^4\,{\rm Z}_{5}^2}{24528751927801}-\frac{40229400\,\alpha_{1}^3\,{\rm Z}_{5}^2}{3323228821}-\frac{237630\,\alpha_{1}^2\,{\rm Z}_{5}^2}{4952651}-\frac{49\,\alpha_{1}\,{\rm Z}_{5}^2}{671}-\frac{364291482375\,\alpha_{1}^4\,{\rm Z}_{5}}{24528751927801}-\frac{130683050\,\alpha_{1}^3\,{\rm Z}_{5}}{3323228821}+\frac{413170\,\alpha_{1}^2\,{\rm Z}_{5}}{4952651}+\frac{97\,\alpha_{1}\,{\rm Z}_{5}}{671}-\frac{956329275125\,\alpha_{1}^4}{24528751927801}-\frac{191385400\,\alpha_{1}^3}{3323228821}+\frac{154815\,\alpha_{1}^2}{4952651}-\frac{53\,\alpha_{1}}{671}-\frac{11}{5}\verb| | $$
$$ \tag{*} \verb|with|\verb| | \\
\left[ \left[ \alpha_{1} , \frac{2315192\,{\rm Z}_{5}^3}{625}+\frac{2328799\,{\rm Z}_{5}^2}{625}+\frac{2266\,{\rm Z}_{5}}{125}+\alpha_{1}^5-\frac{7194704}{3125} \right] , \left[ {\rm Z}_{5} , {\rm Z}_{5}^4+{\rm Z}_{5}^3+{\rm Z}_{5}^2+{\rm Z}_{5}+1 \right] \right] \verb| | $$
$$ \tag{*} \verb|Verification|\verb| | \\
\verb|Numeric values of added radicals|\verb| | \\
\left[ \alpha_{1}=4.17047736063112B-1-1.827644033762094B0\,i , {\rm Z}_{5}=-5.877852522924731B-1\,i-8.090169943749474B-1 \right] \verb| | \\
\verb|Numeric calcuration of the above solutions|\verb| | \\
\left[ -2.236925922272093B-14\,i-3.174929343376513B-1 , -3.608224830031759B-15\,i-1.169169973996244B0 , 3.524958103184872B-15\,i-2.284629676546573B0 , 1.65145674912992B-15\,i-3.309721467890551B0 , 2.070565940925917B-14\,i-3.918985947228983B0 \right] \verb| | \\
\verb|Numeric solutions with allroots(|\verb| |x^5+11\,x^4+44\,x^3+77\,x^2+55\,x+11\verb| |\verb|)|\verb| | \\
\left[ x=-0.3174929343376376 , x=-1.169169973996228 , x=-2.28462967654656 , x=-3.309721467890607 , x=-3.918985947228967 \right] \verb| | $$

 

求めた解を-1/4倍すると元の方程式の解になります。数値的に\( \left(\sin{\frac{n\,\pi}{11}}\right)^{2} \)をn=1,2,3,4,5で計算すると両者が一致することは分かります。

This is a test.
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